1Quasi-homomorphisms

IV Bounded Cohomology

1.3 Poincare translation quasimorphism

We will later spend quite a lot of time studying actions on the circle. Thus, we

are naturally interested in the homeomorphism group of the sphere. We are

mostly interested in orientation-preserving actions only. Thus, we need to define

what it means for a homeomorphism ϕ: S

1

→ S

1

to be orientation-preserving.

The topologist will tell us that ϕ induces a map

ϕ

∗

: H

1

(S

1

, Z) → H

1

(S

1

, Z).

Since the homology group is generated by the fundamental class [

S

1

], invertibility

of

ϕ

∗

implies

ϕ

∗

([

S

1

]) =

±

[

S

1

]. Then we say

ϕ

is orientation-preserving if

ϕ

∗

([S

1

]) = [S

1

].

However, this definition is practically useless if we want to do anything with

it. Instead, we can make use of the following definition:

Definition

(Positively-oriented triple)

.

We say a triple of points

x

1

, x

2

, x

3

∈ S

1

is positively-oriented if they are distinct and ordered as follows:

x

1

x

2

x

3

More formally, recall that there is a natural covering map

π : R → S

1

given by

quotienting by

Z

. Formally, we let

˜x

1

∈ R

be any lift of

x

1

. Then let

˜x

2

, ˜x

3

be

the unique lifts of

x

2

and

x

3

respectively to [

˜x

1

, ˜x

1

+ 1). Then we say

x

1

, x

2

, x

3

are positively-oriented if ˜x

2

< ˜x

3

.

Definition

(Orientation-preserving map)

.

A map

S

1

→ S

1

is orientation-

preserving if it sends positively-oriented triples to positively-oriented triples.We

write

Homeo

+

(

S

1

) for the group of orientation-preserving homeomorphisms of

S

1

.

We can generate a large collection of homeomorphisms of

S

1

as follows — for

any x ∈ R, we define the translation map

T

x

: R → R

y 7→ y + x.

Identifying

S

1

with

R/Z

, we see that this gives a map

T

x

∈ Homeo

+

(

S

1

). Of

course, if n is an integer, then T

x

= T

n+x

.

One can easily see that

Proposition.

Every lift

˜ϕ: R → R

of an orientation preserving homeomorphism

ϕ: S

1

→ S

1

is a monotone increasing homeomorphism of

R

, commuting with

translation by Z, i.e.

˜ϕ ◦ T

m

= T

m

◦ ˜ϕ

for all m ∈ Z.

Conversely, any such map is a lift of an orientation-preserving homeomor-

phism.

We write

Homeo

+

Z

(

R

) for the set of all monotone increasing homeomorphisms

R → R

that commute with

T

m

for all

m ∈ Z

. Then the above proposition says

there is a natural surjection

Homeo

+

Z

(

R

)

→ Homeo

+

(

S

1

). The kernel consists of

the translation-by-

m

maps for

m ∈ Z

. Thus,

Homeo

+

Z

(

R

) is a central extension

of Homeo

+

(S

1

). In other words, we have a short exact sequence

0 Z Homeo

+

Z

(R) Homeo

+

(S

1

) 0

i

p

.

The “central” part in the “central extension” refers to the fact that the image of

Z is in the center of Homeo

+

Z

(R).

Notation.

We write

Rot

for the group of rotations in

Homeo

+

(

S

1

). This

corresponds to the subgroup T

R

⊆ Homeo

+

Z

(R).

From a topological point of view, we can see that

Homeo

+

(

S

1

) retracts to

Rot

. More precisely, if we fix a basepoint

x

0

∈ S

1

, and write

Homeo

+

(

S

1

, x

0

) for

the basepoint preserving maps, then every element in

Homeo

+

(

S

1

) is a product

of an element in

Rot

and one in

Homeo

+

(

S

1

, x

0

). Since

Homeo

+

(

S

1

, x

0

)

∼

=

Homeo

+

([0, 1]) is contractible, it follows that Homeo

+

(S

1

) retracts to Rot.

A bit more fiddling around with the exact sequence above shows that

Homeo

+

Z

(

R

)

→ Homeo

+

(

S

1

) is in fact a universal covering space, and that

π

1

(Homeo

+

(S

1

)) = Z.

Lemma.

The function

F : Homeo

+

Z

(

R

)

→ R

given by

ϕ 7→ ϕ

(0) is a quasi-

homomorphism.

Proof. The commutation property of ϕ reads as follows:

ϕ(x + m) = ϕ(x) + m.

For a real number x ∈ R, we write

x = {x} + [x],

where 0 ≤ {x} < 1 and [x] = 1. Then we have

F (ϕ

1

ϕ

2

) = ϕ

1

(ϕ

2

(0))

= ϕ

1

(ϕ

2

(0))

= ϕ

1

({ϕ

2

(0)} + [ϕ

2

(0)])

= ϕ

1

({ϕ

2

(0)}) + [ϕ

2

(0)]

= ϕ

1

({ϕ

2

(0)}) + ϕ

2

(0) − {ϕ

2

(0)}.

Since 0 ≤ {ϕ

2

(0)} < 1, we know that

ϕ

1

(0) ≤ ϕ

1

({ϕ

2

(0)}) < ϕ

1

(1) = ϕ

1

(0) + 1.

Then we have

ϕ

1

(0) + ϕ

2

(0) − {ϕ

2

(0)} ≤ F (ϕ

1

ϕ

2

) < ϕ

1

(0) + 1 + ϕ

2

(0) − {ϕ

2

(0)}.

So subtracting, we find that

−1 ≤ −{ϕ

2

(0)} ≤ F (ϕ

1

ϕ

2

) − F (ϕ

1

) − F (ϕ

2

) < 1 − {ϕ

2

(0)} ≤ 1.

So we find that

D(f) ≤ 1.

Definition

(Poincare translation quasimorphism)

.

The Poincare translation

quasimorphism T : Homeo

+

Z

(R) → R is the homogenization of F .

It is easily seen that T (T

x

) = x. This allows us to define

Definition

(Rotation number)

.

The rotation number of

ϕ ∈ Homeo

+

(

S

1

) is

T ( ˜ϕ) mod Z ∈ R/Z, where ˜ϕ is a lift of ϕ to Homeo

+

Z

(R).

This rotation number contains a lot of interesting information about the

dynamics of the homeomorphism. For instance, minimal homeomorphisms of

S

1

are conjugate iff they have the same rotation number.

We will see that bounded cohomology allows us to generalize the rotation

number of a homeomorphism into an invariant for any group action.